Solidifying nonmanifold geometry
There are reasons for avoiding non-manifold geometry when modeling. This is one of them. I would not recommend proceeding with this workflow further. It would be best to model the walls with thickness in mind from the beginning.
However, apparently there is an add-on that can help here:
Manifold Solidify

Because of the nature of the operation the directions the walls would get thicker in cannot be chosen. This may introduce some extra calculations needed to plan walls where the space between then is important for example if one is modelling a kitchen one might want to consider standard dimenions of kitchen furniture when planning wall lenghts. However, this might still make work a lot easier in some cases.